It’s probably pointless to argue about the distinction any more, but nark and narc are two different words with different origins and different meanings. A nark (from the Romani word for nose) is a police informant, a civilian who spies on criminals or suspected criminals for the police, regardless of the type of crime. There is also a verb nark, meaning “to act as a police informant.” A narc (clipped form of “narcotics officer”) is a police officer who specializes in investigating drug-related offenses and enforcing drug laws.
I can’t say I recall ever seeing the k-version.
It’s pretty common in British usage.
IMO “nark” is effectively unheard of in US usage.
“Narc” as undercover drug agent or civilian blue-nose do-gooder ratting out drug users and sellers to the conventional police is the one and only early 1960s usage.
By extension now one can “narc” on anyone for anything to any authority. E.g telling somebody about their spouse’s infidelity you’ve seen. " I narced Bob out to Sue over that hookup with Jane."
But IMO now “narc” is becoming quaint. Or I am :).
I leaned it from Shaw’s Pygmalion (a.k.a. My Fair Lady without the songs).
You mean “nark” on anyone 
No, I meant “narc”. Now I may be doubling down on demonstrating my own profound ignorance, but “nark” just isn’t a word in my personal flavor of English. Never seen it written, and would never write it.
Is it a standard word in UK English? Seemingly yes. In US English? If so I utterly missed the memo. Which would not be the first time.
And no, I didn’t Google before writing this post; I’m trying to retain my mental virginity here rather than joining the hivemind of actual US usage, whatever that may be. 
Consider it a form of sincerity. Perhaps misplaced sincerity.
If “nark” is standard in US English, I’ve missed it too. Never seen it written with that spelling and in my lifetime, the meaning has always been the extended one, where you can narc anyone out to anyone for anything. That would include the police for non-drug crimes but isn’t the same as a “snitch”.
Sorry to nitpick, but see bibliophage’s post, about seven posts up. The “informant” meaning may be broader than the “drugs officer” one, but it is not an extension of it. They are two different words (distinct etymologies).
In the UK, they are also spelled differently. In the US, they are spelled the same — but are still different words.
In the US, they are so conflated now that you might argue they’re the same word, as one’s meaning has influenced the other’s. It’s hard for an American to call a drugs officer a “narc” without, on some level, thinking about their potential to rat out the drugs user or dealer. That’s why one meaning feels like an extension of the other — but historically it isn’t, and their semantic spaces were once distinct.
The “extended” usage is not that the “informant” meaning is an extension of the “drugs officer” one. It’s that the informant meaning has been extended from what @bibliophage described
to someone who informs (doesn’t necessarily spy ) someone ( not just police) of someone else’s behavior ( not just crimes ). You could narc me out to the teacher for cheating on a test. “Narc” meaning a narcotics officer is a separate meaning , has a different etymology and may be a different word - but even if you consider them two different words that share both a pronunciation and a meaning ( rather than one word with multiple meanings ) they aren’t confused with each other in US English. It’s not like someone writes or says “narc” but really means “narc” - which is absolutely possible with the different spellings in UK English
Got it! Yes, that is a legit “extension,” for sure.
Not widespread (more common would be “grass”), but understood as criminals’ slang. It certainly features in the New Zealand Brokenwood TV series, where there’s one character who says “I’m no nark” before letting slip some vital fact. There’s also an older usage “Nark it!” meaning something like “Shut up”.